ALMOST PERIODIC SOLUTIONS FOR ABEL EQUATIONS

Using the Liapunov function method, the existence of almost periodic solutions of a 
scalar differential equation is discussed The results for the scalar differential equation are then applied to 
prove the existence and stability of almost periodic solutions of Abel differential equations We obtain 
several interesting results which improve the results due to Chongyou [1] and Dongpin [2].


INTRODUCTION
Many results have been proved for the existence of periodic solutions for the Abel differential equation '(t) (t) + b(t)z + (t) + d(t).
( ) However, attention is hardly given to prove results on the existence of almost periodic solutions of (1 1) The main purpose of this paper is to investigate the existence and stability of most periodic solutions of the Abel equation (1.1).First, we introduce the concept of characteristic function of equation (1 1) and point out that there are relations between the characteristic function and the existence of almost periodic solutions of (1.1)In section 2, we use the Liapunov function method to prove a general theorem on almost periodic solutions of a scalar differential equation.This theorem extends the results of Fink t3], and can be used to simplify the proof of some results ofFink [3].In section 3, the above general theorem is applied to investigate the existence and stability of almost periodic solutions of the Abel differential equations.We also obtain some interesting results which improve the results of Chongyou [1] and Dongpin [2 where f R x R R, is a uniformly almost periodic function in with respect to x E R For any Liapunov function V (t, x) defined on R x R, we define 1 V(2.1)(t, x 1_ {V(t + h,z + hf(t,x)) V(t,x)}.
l conditions of Theorem 2.1 e satisfied, so equation (2 l) possesses most periodic solution (l th mod() C mod(f) LE .1.Suppose conditions of Theorem 2.1 e satisfied The estence of a ufoy cominuous Liapunov nion satisng conditions (i), (ii) of Theorem 2 e iefited by the hull of equation (2.1).Proof ofthis lena is most the same as the one in Fi [4], so we omit it.
Now, we shMl prove that equation (2.l) has d oy h a solution $(t) such that Suppose quation (2 l) has other solution $(t) such that Without loss of generality, one may assume (0)> (0), then (t)> (t) for E R Since b sup (t), there exists a sequence {t,} such that t,--, t' as n oe (t' may be -oe) and we have (t) < ap(t,) _< b.
On the other hand, by (i), we have a(l(t) (t)l) < V(t., (t. ) is bounded and decreasing in t, which implies that the limit lim V(t,x(t) (t)) exists It follows from above that lim V(t,z(t.)(t,)) 0, therefore, lim V(t, p(t) (tx)) O.
Without loss of generality, we consider that To+ and ToTaqb are solutions of equation ( 22) It is easy to verify that hence, To+ TTdp, because equation ( 22) has a unique solution z(t) satisfying a(t) _< z(t) <_ (t).
This implies that (t) is almost periodic with rood(C) E rood(f, a,/3) by making reference to Fink [:3] This completes the proof of Theorem 2
It follows from Theorem 2 that the Abel differential equation (1.1) has an almost periodic solution (t) with mod() C mod(a, b, c, d) Similarly, we can show the second part of the theorem The proof of the theorem is complete From Theorem 3 3, we can get some new results THEOREM 3.4.Suppose b(t) O, a(t), c(t) and d(t) AP(R), and supa(t) < 0, c( then the Abel differential equation '(t) a(t): has an almost periodic solution x(t) with mod(x) C mod(ct, c, d).